Length-scale Effects on Rate-dependent Plasticity

The activation volume V and strain rate sensitivity m of the flow stress provide a direct link between experimentally measurable plastic flow characteristics and underlying operative deformation mechanisms. However, this link can be complicated by such important factors as mobile dislocation density and strain hardening. In the cases of nanopillars or nanospheres the density of mobile dislocations is negligibly small. Under these conditions, the ultra-strength deformation will be rate-limited by the rate at which dislocations are activated from the surface of the sample. In this case the experimentally measured apparent activation parameters are directly related to surface dislocation nucleation [22].

Schuh et al. [28] also measured the probability of occurrence of the first measurable strain jump during nanoindentation tests performed on Pt at temperature up to 200C. The measured rate of strain jumps (𝜂̇), was expressed in terms of an Arrhenius function of the temperature T, applied stress σ, activation energy ΔG and activation volume V of the deformation process as,

𝜂̇ = 𝜂̇₀. 𝑒𝑥𝑝 (Δ𝐺 − 𝜎𝑉

𝑘𝑇 ) (2.22)

The measured ΔG was considerably lower than would be expected for the energy to nucleate a dislocation from a perfect Pt crystal. This suggests that the intrinsic dislocation nucleation was occurring at crystal imperfections within the test sample. Similar conclusions regarding dislocation nucleation stress during nano-indentation have been reached by others [29, 30]. A detailed characterization of the effect of specific types of dislocation nucleation sources on the temperature- and stress-dependence of the rate of intrinsic dislocation nucleation during nano-indentation and micro-pillar compression has yet to be done.

Mook et al. [31] studied freestanding FCC nanoposts of aluminum and permalloy undergoing large strain uniaxial compression tests. The nanoposts were produced, without FIB milling, by e-beam lithography technique with radii ranging from 0.05 to 0.15 m (Figure 2.15) and were nanocrystalline with a (111) texture and surface roughness below 1 nm. They observed a large amount of scatter in the flow stress obtained during compression of these nanoposts and the flow stress tended to increase with decreasing contact radius (Figure 2.15c). The apparent activation volumes V* required for plastic deformation of these structures were found to be quite small when compared to the V* for the deformation of bulk materials, and V* dropped by an order of magnitude as the yield strength increased by a factor of three (Figure 2.15d). They concluded that this is consistent with near-surface dislocation nucleation as the controlling plasticity mechanism.

Figure 2.15: The AFM deflection image of an aluminum nanoposts (a) before compression (initial radius of 82 nm) and (b) after compression (final radius of 175 nm). (c) Flow stress as a function of the effective contact radius and (d) the apparent activation volume versus flow stress, for the aluminum and permalloy nanoposts [31].

(a)

(c)

(b)

Figure 2.16: (a) Flow stress at 10% strain as a function of strain rate for Cu nanopillars (75 to 500 nm) [32]. (b) Activation volume versus diameter at two different strain rates denoting the change in activation volume for the smallest diameters [33].

Jennings [32,33] demonstrated a significant effect of both strain rate and sample size on the compressive strength of single-crystal Cu nanopillars with diameters ranging from 0.075 up to 0.500 µm. The flow stress, at 10% strain, as a function of strain rate for all diameters pillars is shown on Figure 2.16a. The slopes of the curves correspond to the strain rate sensitivity, m ranging between 0.027 and 0.057 displayed an increasing trend with decreasing pillar diameter. The apparent activation volume V* estimated for the two smallest diameter pillars (0.075 and 0.125 µm) was about ~6 and ~7.3b3, but was between 9.6 and 62b3 _{when the pillar diameter was above 0.150 µm (Figure 2.16b). They}

proposed a plasticity mechanism transition from dislocation multiplication via the operation of truncated dislocation sources, also referred to as single-arm sources, in pillars with diameters greater than 0.150 µm to dislocation nucleation from the surface in the smaller diameter samples.

Chen et al. [34] found the V* is about 0.13b3 _{for tensile tests performed on “dislocation}

scarce” single-crystal Pd nanowhiskers. In these samples it was concluded that surface dislocation nucleation was the predominant mechanism controlling plastic yielding.

Schneider et al. [35] performed compression test on BCC [001] and [235] oriented Mo single crystal pillars ranging from 0.2 to 5.0 m diameter. The smaller pillars showed a

strong dependence of flow stress upon loading rate and the calculated V* was between 1.3 to 5.3b3 from the smallest to the largest pillars.

Atomistic modeling of the uniaxial compression on Cu (111) nanowire was studied by Zhu et al. [36]. They created surface steps on the (111) surface and found the calculated

V*s are about 2b3_{. The small magnitude of V* was associated with surface dislocation}

nucleation. They also found that surface sources have a unique kinetic signature: a small activation volume leading to increase strain rate and temperature sensitivities of flow stress. They concluded that the activation volume associated with surface dislocation nucleation is characteristically in the range of 1  10b3.

Figure 2.17: (a) Strain rate sensitivity, m at different temperature for conventional

coarse-grained Ni (circles) and nanocrystalline Ni (squares). (b) Comparison of apparent activation volume at different temperature of nanocrystalline Ni (circles) and conventional coarse-grained Ni (triangles) shows inset in the same plot [37].

Wang et al. [37] carried out strain rate jump tests and stress relaxation tests over a range of deformation temperatures (77–373 K) on electrodeposited nanocrystalline Ni with an average grain size of ~30 nm. The values of the activation volume obtained at different temperatures are much smaller than those for the normal rate-controlling mechanism in bulk FCC metals. This suggests that the thermally activated process in nanocrystalline Ni is different from the conventional forest dislocation cutting mechanism. The strain rate sensitivity, m corresponding to the nanocrystalline Ni was found to be higher than coarse-

grained polycrystalline Ni (Figure 2.17a). A stronger temperature dependence consistent with the higher activation energy and smaller activation volume was observed. There appears to be an additional activated process in action, with a higher activation barrier. The apparent activation volume was found to be about 20b3 for nanocrystalline Ni at about room temperature (Figure 2.17b). This value is one to two orders of magnitude lower than the typical value governing normal dislocation-obstacle controlled plastic deformation of bulk Ni (see inset Figure 2.17b).

Afrin and Ngan [38] carried out compression creep tests at room temperature on micron- sized Ni3Al pillars produced by FIB milling. The nominal creep rates of these

micropillars were found to be very high, at 10-5 s-1. Their observation, shown in Figure 2.18, suggest that the deformation at the pillar head is likely to be caused by surface diffusion. The absence of any sign of deformation within the main body of the pillar indicates that lattice diffusion did not happen, which is reasonable considering the low test temperature. The stress exponent of the creep deformation of these pillars was also close to unity, implying linear diffusional flow being the dominating creep mechanism. Microscopic evidence suggests deformation is due to surface diffusion at the pillar heads. Their estimated activation energy was 0.30  0.02 eV for the low temperature diffusional creep mechanism.

Figure 2.18: SEM images of Ni3Al micropillar (a) before and (b) after creep testing at

2000 µN load. In Fig. b, EDX measurements of the deformed head of the pillar and the shafts revealed that the same chemical compositions, indicating that the material flowing down the pillar heads was indeed Ni3Al. (c) Creep displacement during load hold of the

micropillar and bulk-material [38].

Figure 2.19: The stress-strain response of (a) 6.3 and (b) 0.8 µm, diameter Al pillars [39].

Ng and Ngan [39] studied the overall stress-strain response of micron-sized Al pillars, fabricated by FIB milling, which were subjected to uniaxial compression. They found that the deformation of Al micropillars is a jerky process with a stochastic nature (Figure 2.19). Their TEM investigation revealed that the dislocation density increased only very slightly even after severe plastic deformation. They reported that both the probability of occurrence and the size of the discrete displacement jumps observed during compression of these Al micropillars were exponentially related to the applied stress. When the micropillars were subjected to constant-load creep tests the frequency of the displacement jumps decreased with increasing creep time and a steady strain rate creep component also contributed to the measured creep strain. This suggests that multiple mechanisms of dislocation nucleation controlled and dislocation-obstacle interaction controlled dislocation glide may contribute to the intrinsic creep of sub-micron size metal samples.

Na and Ngan [40] also performed compression testing on FIB milled 6 µm diameter bi- crystal Al pillars. They reported that introducing a grain boundary inside the micropillar resulted in very different mechanical behavior, specifically, the pillar shows much smoother and homogeneous deformation, in addition to a higher work hardening rate, compared to single-crystal pillars of similar size. These improved deformation characteristics were attributed to the increased storage of dislocations as a result of the presence of the grain boundary.

Choi et al. [41] performed room temperature uniaxial creep experiments on nanocrystalline Ni pillars of diameters ranging from 0.6 to 2.0 m. At a given stress, much higher total creep strains and strain rates accrue in the smaller pillars (Figure 2.20), which is likely due to the increased contributions of free surfaces. The calculated apparent activation volume V* increases in the range of 0.5 – 2.2b3 _{with decreasing stress}

and increasing pillar diameter. These measured values of stress exponent and the activation volume suggests that the nanoscale creep event under low stresses may be dominated by diffusion-controlled mechanisms.

Figure 2.20: Creep strain vs. time curve: (a) effect of the applied stress; (b) effect of the

pillar size [41].

Wang et al. [42] conducted both uniaxial creep and nanoindentation creep on nanocrystalline Ni and directly compared the two sets of data. Micropillars, with a nominal diameter of 2 µm, were fabricated by FIB milling and Ni films, with a grain size of 14 nm, were prepared by electrodeposition. The stress exponent under the two test conditions was found to be almost the same, indicating a similar creep mechanism. However, the strain rate measured by nanoindentation creep was about 100 times faster than that by uniaxial creep. The faster creep rate was caused by the facts that the stress state under nanoindentation is more complex and severe than the uniaxial condition. The creep activation energy for the two creeps is also the same and both correspond to the activation energy for grain-boundary diffusion in Ni, suggesting that a similar diffusion- assisted dislocation glide process is operative in both testing situations.